Finite Group Databasebeta

← #29683247

Subgroups — #29683247

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454615C211yesmaximal, characteristic
2#21454459C211yesmaximal, characteristic, complemented
3#21454241C211yesmaximal, characteristic, complemented
4#21446469C211yesmaximal, characteristic, complemented
5#21446418C211yesmaximal, characteristic, complemented
6#21446416C211yesmaximal, characteristic, complemented
7#21446149C211yesmaximal, characteristic
8#21446143C211yesmaximal, characteristic
9#21446097C211yesmaximal, characteristic, complemented
10#21446091C211yesmaximal, characteristic, complemented
11#21445727C211yesmaximal, characteristic, complemented
12#21445631C211yesmaximal, characteristic, complemented
13#21445628C211yesmaximal, characteristic, complemented
14#21445569C211yesmaximal, characteristic, complemented
15C61×#1929C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#41×Dic61C2211yescharacteristic
20#16840230C2211yescharacteristic
21#16839871C411yescharacteristic, complemented
22#16839871C411yescharacteristic, complemented
23#16839866C411yescharacteristic, complemented
24#16839866C411yescharacteristic, complemented
25#16839865C411yescharacteristic, complemented
26#16839865C411yescharacteristic, complemented
27#16839859C411yescharacteristic, complemented
28#16839859C411yescharacteristic, complemented
29#16839820—21no—
30#16839820—21no—
31#16839812—21no—
32#16839812—21no—
33C2×#14519713C2211yescharacteristic
34#16839798C2211yescharacteristic
35#16839790C2211yescharacteristic
36#16839788C2211yescharacteristic
37C2×#14519705C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39C2×#14519703C2211yescharacteristic
40#16839759C2211yescharacteristic
41#16839729C2211yescharacteristic
42#16839719—21no—
43#16839718C2211yescharacteristic
44#16839716—21no—
45#16839714—21no—
46M16×Dic61C2211yescharacteristic
47M16×Dic61C2211yescharacteristic
48C2×#14519687C2211yescharacteristic
49C2×C8×Dic61C2211yescharacteristic
50C4×C8×D61—21no—
51#16839608C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839523C2211yescharacteristic
55#16839522C2211yescharacteristic
56#16839521C2211yescharacteristic
57#16839517C2211yescharacteristic
58#16839516C2211yescharacteristic
59#16839516C2211yescharacteristic
60C2×#108×C61C2211yescharacteristic
61C2×#104×C61C2211yescharacteristic
62C61×#416C2211yescharacteristic
63C61×#408C2211yescharacteristic
64C61×#405C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#336C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519797—22no—
69#14519797—22no—
70C2×#11758202C2×C411yescharacteristic
71C2×#11758200C2×C411yescharacteristic
72C2×#11758201C2×C411yescharacteristic
73C2×#11758199C2×C411yescharacteristic
74#14519785—22no—
75#14519785—22no—
76#14519783—22no—
77#14519783—22no—
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80C2×#11758189—21no—
81C2×#11758189—21no—
82C22×#92418C2311yescharacteristic
83C22×#92418C2311yescharacteristic
84C2×C8×D61—21no—
85C2×C8×D61—21no—
86Q8×Dic61C2×C411yescharacteristic
87Q8×Dic61C2×C411yescharacteristic
88#14519742C2×C411yescharacteristic
89#14519742C2×C411yescharacteristic
90D4×Dic61—21no—
91D4×Dic61—21no—
92D4×Dic61C2×C411yescharacteristic
93D4×Dic61C2×C411yescharacteristic
94#14519739C2×C411yescharacteristic
95#14519739C2×C411yescharacteristic
96#14519737—21no—
97#14519737—21no—
98#14519736—21no—
99#14519736C2311yescharacteristic
100#14519734—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—21no—
105#14519713—22no—
106#14519710C2×C411yescharacteristic
107#14519710C2×C411yescharacteristic
108#14519708C2×C411yescharacteristic
109#14519708C2×C411yescharacteristic
110#14519706—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519705—22no—
115#14519704—22no—
116#14519703—22no—
117#14519702C2×C411yescharacteristic
118#14519702C2×C411yescharacteristic
119#14519699C2×C411yescharacteristic
120#14519699C2×C411yescharacteristic
121C22⋊#92418—21no—
122C22⋊#92418—21no—
123#14519688—21no—
124#14519687—21no—
125#14519687—21no—
126#14519687C2×C411yescharacteristic
127#14519687C2×C411yescharacteristic
128#14519687C2×C411yescharacteristic
129#14519687C2×C411yescharacteristic
130C4×#92418—21no—
131C8×Dic61—21no—
132C8×Dic61—21no—
133C8×Dic61D411yescharacteristic
134C8×Dic61D411yescharacteristic
135C8×Dic61D411yescharacteristic
136C8×Dic61D411yescharacteristic
137#14519684—21no—
138#14519684—21no—
139C2×#41×C61C2311yescharacteristic
140C2×M16×C61C2311yescharacteristic
141C2×M16×C61C2311yescharacteristic
142C22×C8×C61C2311yescharacteristic
143C22×C8×C61C2311yescharacteristic
144#117×C61C2311yescharacteristic
145C2×C42×C61C2311yescharacteristic
146#108×C61—22no—
147#104×C61—22no—
148#103×C61C2×C411yescharacteristic
149#103×C61C2×C411yescharacteristic
150#102×C61C2×C411yescharacteristic
151#102×C61C2×C411yescharacteristic
152C22⋊C8×C61—21no—
153C22⋊C8×C61—21no—
154#97×C61—21no—
155C4×C8×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160C2×Q8×C61C22×C411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C22×C411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61C2411yescharacteristic
166C2×C22⋊D61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61#4111yescharacteristic
171C22×Dic61#4111yescharacteristic
172#11758222—21no—
173#11758222—21no—
174#11758222#4112yes—
175#11758222#4112yes—
176C2×D244C22×C411yescharacteristic
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×Dic122C22×C411yescharacteristic
181M16×C61—21no—
182M16×C61—21no—
183M16×C61—21no—
184M16×C61#4112yes—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61C22×C411yescharacteristic
192C2×C8×C61C22×C411yescharacteristic
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61C22×C411yescharacteristic
196C4⋊C4×C61C22×C411yescharacteristic
197C22⋊C4×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C22⋊Dic61—21no—
202C22⋊Dic61—21no—
203C22⋊Dic61—21no—
204#11758202—22no—
205#11758201—22no—
206#11758200—22no—
207#11758199—22no—
208#11758198—21no—
209#11758198—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C22×C411yescharacteristic
213C4⋊Dic61C22×C411yescharacteristic
214#11758196—21no—
215#11758196—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227#11758194—21no—
228#11758194—21no—
229#11758194—21no—
230#11758194#4112yes—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418C22×C411yescharacteristic
238C2×#92418C22×C411yescharacteristic
239C2×#92418C2×D411yescharacteristic
240C2×#92418C2×D411yescharacteristic
241#11758189—24no—
242#11758189—24no—
243C8×D61—24no—
244C8×D61—24no—
245C23×C61—21no—
246C23×C61C2×#4111yescharacteristic
247C22×D61—21no—
248Q8×C61—21no—
249Q8×C61—21no—
250D4×C61—21no—
251D4×C61—21no—
252D4×C61—41no—
253D4×C61—41no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—22no—
269C22⋊D61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61C4×D411yescharacteristic
281C2×Dic61C4×D411yescharacteristic
282C2×Dic61C4×D411yescharacteristic
283C2×Dic61C4×D411yescharacteristic
284D244—21no—
285D244#11712yes—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—42no—
289C4×D61—42no—
290Dic122—21no—
291Dic122#11712yes—
292C8×C61—21no—
293C8×C61—21no—
294C8×C61—21no—
295C8×C61—22no—
296C8×C61—21no—
297C8×C61—41no—
298#92418—21no—
299#92418—21no—
300#92418—21no—
301#92418—22no—
302#92418—21no—
303#92418—41no—
304#21138—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C22×C61#52911yescharacteristic
312C22×C61#52911yescharacteristic
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61C2×#11711yescharacteristic
321C4×C61C4×#4111yescharacteristic
322C4×C61#52511yescharacteristic
323C4×C61#52511yescharacteristic
324Dic61—21no—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—22no—
329Dic61—41no—
330Dic61—41no—
331Dic61—81no—
332#2943—611no—
333#2922—611no—
334C4×#131—611no—
335#1929—611no—
336#1929D6111yescharacteristic, complemented
337#1922—611no—
338#1922—611no—
339#1876—611no—
340#1876—611no—
341#1866—611no—
342#1866—611no—
343#1778—611no—
344#1759—611no—
345#1759—611no—
346#1746—611no—
347C2×C61—21no—
348C2×C61—21no—
349C2×C61—81no—
350C2×C61#229011yescharacteristic
351C2×C61#295511yescharacteristic
352C2×C61#295511yescharacteristic
353D61—42no—
354C2×#135—611no—
355C2×#131—611no—
356C2×#131—611no—
357C4×#41—611no—
358C4×#41—611no—
359#442—611no—
360#442—611no—
361#441—611no—
362#441—611no—
363#440—611no—
364#440—611no—
365#440—611no—
366#440—611no—
367#433—1221no—
368#433—1221no—
369#431—1221no—
370#431—1221no—
371C2×#108—611no—
372C2×#108C2×D6111yescharacteristic
373#427—611no—
374#424—611no—
375#424—611no—
376C2×#104—611no—
377C2×#104—611no—
378C2×#104—611no—
379C2×#104C2×D6111yescharacteristic
380#416—611no—
381#416C2×D6111yescharacteristic
382#408—611no—
383#408C2×D6111yescharacteristic
384#405—611no—
385#405—1221no—
386#405—1221no—
387#405C2×D6111yescharacteristic
388C4×M16—611no—
389C4×M16—611no—
390C4×M16—1221no—
391C2×#97—611no—
392C2×C4×C8—611no—
393C2×C4×C8—1221no—
394#376—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#339—611no—
400#339C2×D6111yescharacteristic
401#336—611no—
402#336—611no—
403#336—611no—
404#336C2×D6111yescharacteristic
405C61#2113811yescharacteristic, complemented, Sylow
406C2×#41—611no—
407C2×#41C22×D6111yescharacteristic
408#135—1222no—
409#135—1222no—
410C2×Q16—611no—
411C2×SD16—611no—
412C2×SD16—611no—
413C2×D8—611no—
414#131—1222no—
415#131—1222no—
416#131—1222no—
417#131—1222no—
418C2×M16—611no—
419C2×M16—611no—
420C2×M16—1221no—
421C2×M16—1221no—
422C2×M16C22×D6111yescharacteristic
423C2×M16C22×D6111yescharacteristic
424C22×C8—611no—
425C22×C8—611no—
426C22×C8—1221no—
427C22×C8—1221no—
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430C4×Q8—611no—
431C4×Q8—611no—
432C4×Q8—611no—
433C4×Q8—611no—
434C4×D4—611no—
435C4×D4—611no—
436C4×D4—611no—
437C4×D4—611no—
438C4×D4—1221no—
439C4×D4—1221no—
440C4×D4—1221no—
441C4×D4—1221no—
442#117—611no—
443#117—1221no—
444#117—1221no—
445#117C22×D6111yescharacteristic
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#108—22no—
452#108—1222no—
453#107—611no—
454#107—611no—
455#106—611no—
456#106—611no—
457C4⋊C8—1221no—
458C4⋊C8—1221no—
459C4⋊C8—1221no—
460C4⋊C8—1221no—
461#104—22no—
462#104—1222no—
463#104—1222no—
464#104—1222no—
465#103—611no—
466#103—611no—
467#103C4×D6111yescharacteristic
468#103C4×D6111yescharacteristic
469#102—611no—
470#102—611no—
471#102C4×D6111yescharacteristic
472#102C4×D6111yescharacteristic
473C22⋊C8—21no—
474C22⋊C8—21no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—21no—
478#97—611no—
479#97—611no—
480#97—611no—
481#97—611no—
482#97—1221no—
483#97—1221no—
484#97—1221no—
485C4×C8—21no—
486C4×C8—611no—
487C4×C8—611no—
488C4×C8—611no—
489C4×C8—611no—
490C4×C8—1221no—
491C4×C8—1221no—
492C4×C8—1221no—
493#95—1221no—
494#95—1221no—
495#41—21no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—612no—
500#41—612no—
501#41—1221no—
502#41—1221no—
503C2×Q8—611no—
504C2×Q8C2×C4×D6111yescharacteristic
505C2×D4—21no—
506C2×D4—611no—
507C2×D4—1221no—
508C2×D4C2×C4×D6111yescharacteristic
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4C23×D6111yescharacteristic
520Q16—1222no—
521SD16—1222no—
522SD16—1222no—
523D8—1222no—
524M16—21no—
525M16—21no—
526M16—21no—
527M16—612no—
528M16—1224no—
529M16—1224no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16#1175822412yes—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—611no—
541C2×C8—611no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—1224no—
545C2×C8—1224no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8C2×C4×D6111yescharacteristic
553C2×C8C2×C4×D6111yescharacteristic
554C2×C8D4×D6111yescharacteristic
555C2×C8D4×D6111yescharacteristic
556C4⋊C4—611no—
557C4⋊C4—611no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4C2×C4×D6111yescharacteristic
563C4⋊C4C2×C4×D6111yescharacteristic
564C22⋊C4—21no—
565C22⋊C4—1221no—
566C22⋊C4—1221no—
567C22⋊C4—1221no—
568C22⋊C4—1221no—
569C22⋊C4—1221no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—1221no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—1221no—
583C42—1221no—
584C23—21no—
585C23—1221no—
586C23#41×D6111yescharacteristic
587Q8—21no—
588Q8—21no—
589Q8—612no—
590Q8—1221no—
591D4—21no—
592D4—21no—
593D4—41no—
594D4—41no—
595D4—612no—
596D4—1222no—
597D4—1222no—
598D4—1221no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4C22×C4×D6111yescharacteristic
626C2×C4C2×D4×D6111yescharacteristic
627C2×C4#41×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4C2×#1175822411yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—21no—
635C8—22no—
636C8—41no—
637C8—1221no—
638C8—1221no—
639C8—1222no—
640C8—1221no—
641C8—1221no—
642C8—2441no—
643C22—21no—
644C22—21no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22C4×D4×D6111yescharacteristic
652C22#1684029111yescharacteristic
653C22#1684029511yescharacteristic
654C4—41no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—1222no—
659C4—1221no—
660C4—1221no—
661C4—1221no—
662C4—1221no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4#117×D6111yescharacteristic
667C4C4×#1175822411yescharacteristic
668C4#1684027011yescharacteristic
669C4#1684027211yescharacteristic
670C2—21no—
671C2—21no—
672C2—81no—
673C2—2442no—
674C2#2144826111yescharacteristic
675C2#2145471711yescharacteristic
676C2#2145471911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1929C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1929D61
hypocenterC61#21138
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1929

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454459C2semidirect
3. #21454241C2semidirect
4. #21446469C2semidirect
5. #21446418C2semidirect
6. #21446416C2semidirect
9. #21446097C2semidirect
10. #21446091C2semidirect
11. #21445727C2semidirect
12. #21445631C2semidirect
13. #21445628C2semidirect
14. #21445569C2semidirect
15. C61×#1929C2semidirect
21. #16839871C4semidirect
22. #16839871C4semidirect
23. #16839866C4semidirect
24. #16839866C4semidirect
25. #16839865C4semidirect
26. #16839865C4semidirect
27. #16839859C4semidirect
28. #16839859C4semidirect
336. #1929D61semidirect
405. C61#21138semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214546152
#214544592
#214542412
#214464692
#214464182
#214464162
#214461492
#214461432
#214460972
#214460912
#214457272
#214456312
#214456282
#214455692
C61×#19292
#2113861